2015/03/09 by Tord Sjödin, Sjödin, Tord
Mathematics · #FOS: Mathematics #Primary 60E10 Secondary 62H05 #Probability (math.PR) #math.PR #msc:60E10 #msc:62H05
paper · pdf · doi:10.48550/arxiv.1503.02491
13 pages
arxiv created 2015/03/09 · arxiv updated 2015/03/10
The class HCM consists of all nonnegative functions f such that f(uv)*f(u/v)is completely monotone with respect to w=v+1/v, for all fixed positive numbers u, and has been extensively studied for a long time. It is closed with respect to (wr) a number of useful operations such as products and quotients of independent random variables, some changes of variables and the Laplace transform. We consider its multivariate (bivariateI counterparts MVHCM (BVHCM) and study some of their properties. In particular, we prove that MVHCM is closed wrt the Laplace transform and use this to define a class BVHCM-L of bivariate random vectors having this property. Then BVHCM-L contains Bondesson's class of random vectors in BVHCM in the strong sense. Finally, we show that BVHCM, in contrast to HCM, is not closed wrt multiplication of independent bivariate random vectors.